1. Created by the a.m.r.a.h.y group

2. first topic: number problems Of all the word problems, the number problems are the easiest to translate into equations since the relationships among the numbers are directly stated in the problem.

3. ILLUSTRATIVE EXAMPLE One number is two more than thrice another. Their sum is 30. Find these numbers. Solution: READ: Reading the problem thoroughly, we know two things about the numbers. a a) their sizes : one of them is two more than thrice the other and b b) their sum: the sum is 30. REPRESENT: If we represent the numbers using the first sentence we have: l let x= first number t then: 3x+2= other number

4. R RELATE : The relationship between the number x and 3x+2 and the other number,30, gives as an equation. E EQUATE: X X+3X+2= 30 S SOLVE: In solving the equation we have: x+3x+2=30 4x+2=30 4x=28 x=7 Therefore, the first number is 7 and the other number is 3x+2=3(7)+2=23 Answer :7 &23 Prove: a)T heir sum is 30: 7+23=30 b) 2 3 is two more than thrice 7: 3(7)+2=21+2=23

5. Exercises 1.) The sum of two numbers is 29 and their difference is 6. find the numbers. 2.) the smaller of two numbers is thrice the larger. The larger number is eight more than the smaller one. Find the numbers.

6. Solution: 1.) x-(29-x)=5 x-29+x=5 2x-29=5 2x=34 x=17 Other number is 29-x=29-17=12 2.) x=3x+8 3x+8=x 2x=-8 x=-4 3x=-12 The answers are: -4 & -12

7. SECOND TOPIC: ODD, EVEN, AND CONSECUTIVE INTEGERS The word consecutive means following in order without interruption. Even numbers are whole numbers divisible by 2 while whole numbers which are not divisible by 2 are odd numbers. Consecutive even integers are even numbers in uninterrupted order which is the same as when you count by twos such as 2,4,6,8,&10. Consecutive odd integers are odd numbers in uninterrupted order, such as 7,9,11,13,& 15.

8. Illustrative example: 1.) The sum of three consecutive integers is 90. Find the integers. solution: let x= first number then x+1= next consecutive integer. and x+2= third consecutive integer. their sum is 90 manipulating the equation we have: x+(x+1)+(x+2)= 90 x+x+1+x+2=90

9. 3x+3=90 3x+3=90 3x= 87 3x= 87 x=29 1 st integer x=29 1 st integer x+x1= 2 nd integer x+x1= 2 nd integer x+2= 3 rd integer x+2= 3 rd integer answer: the consecutive integers are 29,30&31 answer: the consecutive integers are 29,30&31 Prove: 29+30+31=90

10. exercises: 1.F ind two consecutive integer such that if we triple the first and double the second, the sum is 77. 2. the sides of a triangle are consecutive integers. If the perimeter of this triangle is 192 cm, find the length of each sides.

11. solution: 1.3 (x)+2(x+1)=77 3x+2x+2=77 5x+2= 75 x=15 1st integer x+1= 16 2nd integer Answer: the consecutive integers are 15 & 16 Proof: triple 15: 45 double 16:32 total:77 2. x+x+1+x+2=192 3x+3= 192 3x= 189 x=63 cm x+1= 64 cm x+2= 65 cm answer: the lengths of the sides of a triangle are 63,64,65 cm.

13. The delineation of a sphere on a flat plane is similar to painting. For just as the painters seek to imitate objects exactly...geometricians and astronomers delineate on a flat plane solid objects such as octahedrons and cubes and all spherical bodies, like the stars, the heavens, and the earth.

14. If all the arts aspire to the condition of music, all the sciences aspire to the condition of mathematics.

15. Let no one ignorant of mathematics enter here.

16. As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.

17. No, it is a very interesting number, it is the smallest number expressible as a sum of two cubes in two different ways.

18. There is no "royal road" to geometry.